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As the title suggests, does anyone have recommendations for learning EM with differential forms? Both undergraduate and graduate level textbook suggestions are welcomed. I know there's Lindell's Differential Forms in Electromagnetics and Hehl's Foundations of Classical Electrodynamics, but am struggling a bit with the notations still. Any other recommendations or suggestions to tackle the topic?

In addition, are there some good EM book with a focus on condensed matter that uses differential forms?

Andrew Li
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(These are from my comments written while awaiting the re-opening.)

Resources for differential forms in electrodynamics [in addition to Lindell and Hehl].
(In addition, consult the references within them):

  • Warnick - Consider Electromagnetics scholarsarchive.byu.edu/facpub/1314 and Teaching electromagnetic field theory using differential forms scholarsarchive.byu.edu/facpub/669 by Warnick, Selfridge, Arnold. The first document looks like part of a textbook intended for undergraduates. It develops electromagnetism using differential forms, with emphasis on visualization in the spirit of Misner-Thorne-Wheeler and Bill Burke (likely inspired by visualizations by Schouten).
    See also Differential Forms and Electromagnetic Field Theory (2006) by Warnick and Russer https://www.jpier.org/PIER/pier148/09.14063009.pdf

  • Burke - Applied Differential Geometry by William Burke has sections on electromagnetism using differential forms, with emphasis on visualization. See also Burke's Spacetime, Geometry, Cosmology. (It's fair to say that Burke inspired Warnick.) See also Burke's unpublished incomplete set of notes: Div, Grad, and Curl are Dead people.ucsc.edu/~rmont/papers/Burke_DivGradCurl.pdf

  • Bamberg & Sternberg - A Course in Mathematics for Students of Physics by Bamberg and Sternberg develop electromagnetism using differential forms in 3D-space and in (3+1)-D spacetime. They also develop classical thermodynamics using differential forms. However, this text is rather mathematical. It's not for the typical physics undergraduate.

  • Bossavit - "On the geometry of electromagnetism" by Alain Bossavit https://www.researchgate.net/publication/254470625_On_the_geometry_of_electromagnetism gives a mathematical presentation of electromagnetism that (in my opinion) fills in some of the details in Burke's presentation. Consider other articles by Bossavit researchgate.net/profile/Alain-Bossavit

  • Ingarden & JamioĊ‚kowski - Classical Electrodynamics (1985) by Roman S. Ingarden and Andrzej JamioĊ‚kowski is a textbook that uses differential forms, as well as tensor notation (including the Hodge-dual and tensor densities) in space and spacetime. The treatment is rather mathematical. It is appropriate for mathematically-advanced undergraduates.

  • Schouten & van Dantzig - I think visualization of the differential forms goes back to Schouten's Ricci Calculus (1924) (see p. 15 and 33 in Der Ricci-Kalkul http://resolver.sub.uni-goettingen.de/purl?PPN373339186 for a diagram of 1-forms [covectors]). See also page 22 for vectors and pseudovectors.

    van Dantzig (an assistant to Schouten) had some interesting papers emphasizing the pre-metric viewpoint. Van Dantzig, D. (1934). "The fundamental equations of electromagnetism, independent of metrical geometry." Mathematical Proceedings of the Cambridge Philosophical Society, 30(4), 421-427. https://doi.org/10.1017/S0305004100012664

    van Dantzig has some useful diagrams for differential forms in "On the geometrical representation of elementary physical objects and the relations between geometry and physics" : (Nieuw Archief voor Wiskunde 3e serie, 2(1954), p 73-89) https://ir.cwi.nl/pub/8429

    They published "On ordinary quantities and W-quantities. Classification and geometrical applications" Compositio Mathematica, tome 7 (1940), p. 447-473 ( http://www.numdam.org/item/CM_1940__7__447_0.pdf ) which deals with even and odd (ordinary and twisted [following Burke]) quantities.
    See Schouten's Tensor Analysis for Physicists for a readable but terse presentation on these ideas.

  • Jancewicz - has published on aspects of visualizing and intepreting differential forms in electromagnetism: https://scholar.google.com/citations?user=Qm_PnggAAAAJ&hl=pl

  • Discrete Exterior Calculus - has been used in computer graphics and computational electromagnetism. This is another way to get a handle on differential forms. Follow the references.

(I have an interest in the visualization and physical interpretation of differential forms in electrodynamics in space and in spacetime. My interest also extends to relativity, thermodynamics, and mechanics.)

robphy
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